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%I #55 Dec 17 2023 16:33:16
%S 1,1,2,6,24,40,720,5040,4480,362880,3628800,13305600,479001600,
%T 6227020800,29059430400,1307674368000,1609445376000,13173608448000,
%U 6402373705728000,121645100408832000,810967336058880000,4644631106519040000,86461594444431360000
%N Denominators of the coefficients in exp(x/(1-x)) power series.
%C Define c(n)=A067764(n)/A067653(n). For a given sequence s(n) consider P[s(n)](z):=e^(-z/(1-z))*Sum_{k>=0} s(k)c(k)z^k. Regarding complex valued abelian limitation the following holds true: if s(n) is convergent (to the limit s) then lim P[s(n)](z)=s as z tends to +1 in a certain subdomain D of the unit circle. There are two constraints: (1) D contains the line [0,1[. (2) There is a d>0 such that the intersection of {w|Re(w)>1-d} and D is a nonempty subset of a generalized Stolz set defined by {w||Im(w)|<=t*(1-Re(w))^(3/2)}, t<1. If z tends to +1 from outside such a domain that limit doesn't exist in general. - _Hieronymus Fischer_, Oct 20 2010
%C There is a significant overlap between the terms of this sequence and the terms of factorials (A000142): 44 of the first 100 terms of this sequence are also factorials. - _Harvey P. Dale_, Oct 27 2011
%C If there is no cancellation in the defining sum, a(n) = n!. In particular, a(p) = p! for prime p. In any case a(n) | n!. - _Charles R Greathouse IV_, Oct 27 2011
%D H. Fischer, Eine Theorie komplexwertiger Abelscher Limitierungsmethoden (A theory of complex valued abelian limitation methods), Dissertation (1987), pp. 29-32.
%D K. Knopp, Theory and application of infinite series, Dover, p. 547.
%D O. Perron, Über das infinitäre Verhalten der Koeffizienten einer gewissen Potenzreihe, Archiv d. Math. u. Phys. (3), Vol. 22, pp. 329-340, 1914.
%D K. Zeller, W. Beekmann, Theorie der Limitierungsverfahren, Springer-Verlag, Berlin (1970).
%H Alois P. Heinz, <a href="/A067653/b067653.txt">Table of n, a(n) for n = 0..450</a>
%H D. Borwein, <a href="http://dx.doi.org/10.1017/S0080454100007561">On methods of summability based on power series</a>, Proc. Royal Soc. Edinburgh, Sect. A Volume 64 / Issue 04 / January 1957, pp 342-349.
%H Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%F a(n) is the denominator of Sum_{i=1..n} binomial(n-1, i-1)/i!.
%e exp(x/(1-x)) = 1 + x + (3/2)*x^2 + (13/6)*x^3 + (73/24)*x^4 + (167/40)*x^5 + (4051/720)*x^6 + (37633/5040)*x^7 + (43817/4480)*x^8 + (4596553/362880)*x^9 + ... .
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add((n-k)*b(k), k=0..n-1)/n)
%p end:
%p a:= n-> denom(b(n)):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 12 2016
%t Denominator[Rest[CoefficientList[Series[Exp[x/(1-x)],{x,0,20}],x]]] (* _Harvey P. Dale_, Oct 26 2011 *)
%t r[n_] := If[n == 0, 1, Hypergeometric1F1[1 - n, 2, -1]]; Table[Denominator@ r[n], {n, 0, 22}] (* _Peter Luschny_, Feb 02 2019 *)
%o (PARI) apply(x->denominator(x),Vec(exp(x/(1-x)))) \\ _Charles R Greathouse IV_, Oct 27 2011
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-x)))); [Denominators(b[n]): n in [1..m]]; // _G. C. Greubel_, Dec 04 2018
%o (SageMath) [denominator(sum(binomial(n-1, j-1)/factorial(j) for j in (1..n))) for n in range(30)] # _G. C. Greubel_, Dec 04 2018
%Y Cf. A067764.
%K nonn,frac
%O 0,3
%A _Benoit Cloitre_, Feb 03 2002
%E a(0)=1 prepended by _Alois P. Heinz_, May 12 2016
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